Question

Use the table to find the residual points.

A 4-column table with 5 rows. The first column is labeled x with entries 1, 2, 3, 4, 5. The second column is labeled given with entries negative 0.7, 2.3, 4.1, 7.2, 8. The third column is labeled predicted value with entries negative 0.28, 1.95, 4.18, 6.41, 8.64. The fourth column is labeled residual value with all entries blank.

Which residual plot is the correct one for the data?

A graph shows the horizontal axis numbered negative 1 to positive 7 and the vertical axis numbered negative 1.5 to positive 1.5. Points are scattered all over the graph.
A graph shows the horizontal axis numbered negative 1 to positive 7 and the vertical axis numbered negative 1.5 to positive 1.5. Points are scattered all over the graph.
A graph shows the horizontal axis numbered negative 1 to positive 7 and the vertical axis numbered negative 1.5 to positive 1.5. Points are scattered all over the graph.
A graph shows the horizontal axis numbered negative 1 to positive 7 and the vertical axis numbered negative 1.5 to positive 1.5. Points are scattered all over the graph.

Answer 1

To find the residual points, we first need to calculate the residual for each entry in the table. The residual is calculated as follows:

\[
\text{Residual} = \text{Given Value} - \text{Predicted Value}
\]

Now, let's calculate the residuals for each \( x \):

1. For \( x = 1 \):
Given = -0.7, Predicted = -0.28
Residual = -0.7 - (-0.28) = -0.7 + 0.28 = -0.42

2. For \( x = 2 \):
Given = 2.3, Predicted = 1.95
Residual = 2.3 - 1.95 = 0.35

3. For \( x = 3 \):
Given = 4.1, Predicted = 4.18
Residual = 4.1 - 4.18 = -0.08

4. For \( x = 4 \):
Given = 7.2, Predicted = 6.41
Residual = 7.2 - 6.41 = 0.79

5. For \( x = 5 \):
Given = 8, Predicted = 8.64
Residual = 8 - 8.64 = -0.64

Now the residual values are:

| \( x \) | Given | Predicted Value | Residual Value |
|---------|-------|------------------|------------------|
| 1 | -0.7 | -0.28 | -0.42 |
| 2 | 2.3 | 1.95 | 0.35 |
| 3 | 4.1 | 4.18 | -0.08 |
| 4 | 7.2 | 6.41 | 0.79 |
| 5 | 8 | 8.64 | -0.64 |

Now, the residual values are:
- For \( x = 1 \): -0.42
- For \( x = 2 \): 0.35
- For \( x = 3 \): -0.08
- For \( x = 4 \): 0.79
- For \( x = 5 \): -0.64

To plot these on a residual plot:
- The x-coordinates will be the values of \( x \) (1, 2, 3, 4, 5).
- The y-coordinates will be the corresponding residual values (-0.42, 0.35, -0.08, 0.79, -0.64).

Now you would need to identify which of the given graphs reflects this distribution of residuals. The correct residual plot will show points at the following coordinates:
- \( (1, -0.42) \)
- \( (2, 0.35) \)
- \( (3, -0.08) \)
- \( (4, 0.79) \)
- \( (5, -0.64) \)

Make sure to look for a plot where the points do not form any clear pattern but are scattered vertically around the horizontal axis (y=0). This randomness and scattering around zero are typical characteristics of a good residual plot.